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Speckle interferometry with adaptive optics corrected solar data

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Speckle interferometry with adaptive optics corrected solar data Powered By Docstoc
					A&A 488, 375–381 (2008)                                                                                             Astronomy
DOI: 10.1051/0004-6361:200809894                                                                                     &
c ESO 2008                                                                                                          Astrophysics


   Speckle interferometry with adaptive optics corrected solar data
                                          F. Wöger1 , O. von der Lühe2 , and K. Reardon1 ,3

     1
         National Solar Observatory, PO Box 62, Sunspot, NM 88349, USA
         e-mail: fwoeger@nso.edu
     2
         Kiepenheuer-Institut für Sonnenphysik, Schöneckstr. 6, 79104 Freiburg, Germany
     3
         INAF – Osservatorio Astrosico di Arcetri, 50125 Firenze, Italy
     Received 2 April 2008 / Accepted 17 May 2008

                                                                 ABSTRACT

     Context. Adaptive optics systems are used on several advanced solar telescopes to enhance the spatial resolution of the recorded data.
     In all cases, the correction remains only partial, requiring post-facto image reconstruction techniques such as speckle interferometry
     to achieve consistent, near-diffraction limited resolution.
     Aims. This study investigates the reconstruction properties of the Kiepenheuer-Institut Speckle Interferometry Package (KISIP) code,
     with focus on its phase reconstruction capabilities and photometric accuracy. In addition, we analyze its suitability for real-time re-
     construction.
     Methods. We evaluate the KISIP program with respect to its scalability and the convergence of the implemented algorithms with
     dependence on several parameters, such as atmospheric conditions. To test the photometric accuracy of the final reconstruction, com-
     parisons are made between simultaneous observations of the Sun using the ground-based Dunn Solar Telescope and the space-based
     Hinode/SOT telescope.
     Results. The analysis shows that near real-time image reconstruction with high photometric accuracy of ground-based solar observa-
     tions is possible, even for observations in which an adaptive optics system was utilized to obtain the speckle data.
     Key words. techniques: high angular resolution – techniques: image processing – techniques: interferometric – sun: photosphere



1. Introduction                                                           the data amount is reduced by a factor of around 100. Thus, the
                                                                          possibility of real-time data reduction becomes attractive when
Adaptive optics (AO) systems have been introduced to many                 post-processing techniques like speckle interferometry are con-
solar telescopes in the recent years, making large aperture fa-           sidered for image reconstruction. Some of the aspects of the ap-
cilities feasible. However, it is evident that any AO correction          plication of post-processing algorithms to speckle data in near
is only partial. Thus, to achieve diffraction limited performance          real-time have already been explored by Denker et al. (2001).
of the telescope, further post-processing of the observations be-             In this article, we present the characteristics of the
comes necessary. Several algorithms for image reconstruction              Kiepenheuer-Institut Speckle Interferometry Package (KISIP,
have evolved as computational power has increased rapidly. On             von der Lühe 1993; Mikurda & von der Lühe 2006), which has
the one hand, techniques based on blind deconvolution like mul-           been rewritten in the C programming language and enhanced
tiframe blind deconvolution or the even more general multi-               for parallel processing1. In Sect. 2, we give an overview of the
object multiframe blind deconvolution have evolved and be-                implemented algorithms. Section 3 describes our study of the
come popular in the recent years (van Noort et al. 2005). On              performance of the two implemented phase reconstruction al-
the other hand, techniques based on speckle interferometry that           gorithms, as well as the overall scalability of the code with an
have evolved since the mid-1970s have been refined and im-                 increasing number of computational nodes. In addition, the pho-
proved (Labeyrie 1970; Knox & Thompson 1974; Weigelt 1979;                tometric accuracy of the final reconstruction is tested with both a
Lohmann et al. 1983) during the 1980s.                                    ground- and space-based telescope co-temporally observing the
    The rapid development of computer technology, especially              same target on the Sun.
in the field of multi-core processors, makes a real-time appli-
cation of reconstruction algorithms to speckle interferometric
data feasible and warrants further development. The need for              2. Algorithmic details
real-time – or at least near real-time – processing is clear when
considering that speckle data is observed at high data rates:             In this section, we briefly describe the internal details of how
in general, a single “speckle burst” consists of approximately            KISIP was implemented to allow for parallel processing as well
100 images observed at a frame rate of around 15 images per sec-          as the employed algorithms used for the reconstruction of the
ond (or higher). When observing several hours a day this leads            Fourier phase and amplitude.
to a data volume of several hundred gigabytes of unreduced data               In general, the imaging process through atmosphere and
per day. Even though the cost per byte is continually decreasing,         telescope is best described in the Fourier domain. Using the
the handling (transfer and distribution) is a costly and lengthy
process. Thus, the reduction of speckle data at the telescope site        1
                                                                            The full C sources of the package are available at https://forge.
is an important step to increase the telescope’s efficiency because         kis.uni-freiburg.de/kisip


                                                    Article published by EDP Sciences
376                                   F. Wöger et al.: Speckle interferometry with adaptive optics corrected solar data

incoherent, space-invariant imaging equation, we get for a
speckle burst consisting of N images
Ii ( f ) = O( f ) Si ( f ),   1 ≤ i ≤ N,                                 (1)
where f is the two-dimensional, spatial frequency and Ii ( f ) is
the Fourier transform of the ith observed image of the object de-
scribed by O( f ). The term Si ( f ) is the ith transfer function that
incorporates aberrations due to both atmosphere and telescope,
and is generally a complex function. We assume that there are
no static aberrations in the telescope, thus complex contribu-
tions to the transfer function only arise from phase distortions
due to Earth’s turbulent atmosphere. This is justified by the fact
that most solar telescopes today use AO systems. These systems
are capable to correct some of the atmospheric aberrations in                   Fig. 1. Time used for one reconstruction versus numbers of computation
real-time, and additionally remove most of the static aberrations               nodes used. Either 212992 cross-spectrum (KT) or 221320 bispectrum
efficiently.                                                                      (IWLS) values for averaging.
    At an early stage of the reconstruction process, each recorded
short-exposed frame is split into subframes that have roughly the
size as the isoplanatic patch (the area in the field of view for                     The other algorithm available within the package is a speckle
which the optical transfer function is considered constant) and                 masking (or triple correlation) algorithm. Speckle masking al-
that overlap by half of their size. This makes a parallel treatment             gorithms, the generalization of Knox and Thompson’s idea us-
of the subframes easy as they are sent to separate computation                  ing the bispectrum, have been used since Weigelt (1979) and
nodes using the message passing interface (MPI Forum 1997).                     Lohmann et al. (1983). The bispectrum is defined as
The KISIP package separates the image’s Fourier phases from its                 B(u, u) = Ii (u) Ii (u) Ii∗ (u + u) i
amplitudes. The Fourier phases are treated with unity amplitude                         = Oi (u) Oi (u) O∗ (u + u)
                                                                                                             i                                    (4)
by both of the implemented phase reconstruction algorithms,                               × S i (u) S i (u) S i∗ (u + u) i .
which are described in further detail below. Fourier amplitudes
are reconstructed independently. In what follows, we give a brief               In analogy to Eq. (3), it can be shown that the speckle masking
overview over these well-known techniques that form the basis                   transfer function
of KISIP.                                                                       S MT F( f ) = S i (u) S i (u) S i∗ (u + u)                        (5)
                                                                                                                              i

                                                                                is a real valued entity and remains finite up to the diffraction
2.1. Phase reconstruction                                                       limit (von der Lühe 1985). The rather stringent restriction for
The KISIP program incorporates two different algorithms to                       δ’s magnitude in the extended KT approach is relaxed for u and
reconstruct the object’s Fourier phases.                                        u from the seeing to the diffraction limit. The algorithm imple-
    In one case, the package uses an extension of the Knox-                     mented in KISIP is based on a technique described by Matson
Thompson (KT) algorithm (Knox & Thompson 1974) which is                         (1991), who proposes the reconstruction of phases using an iter-
based on the original authors’ idea to use average cross-spectra                ative weighted least-squares (IWLS) fit to the bispectrum. Thus,
for the reconstruction of the object’s Fourier phases. The Knox-                it makes full use of the bispectrum that was computed with user
Thompson average cross-spectrum is defined as                                    specified truncation parameters to restrict it to a manageable size
                                                                                (e.g., Pehlemann & von der Lühe 1989).
C( f , δ) = Ii ( f ) Ii∗ ( f − δ) i                                                  As the extended KT algorithm uses cross-spectra, i.e. mul-
                                                                         (2)
          = O( f ) O∗ ( f − δ) S i ( f ) S i∗ ( f − δ) i .                      tiplications of two Fourier phase values (see Eq. (2)), it is com-
                                                                                putationally less expensive than a speckle masking algorithm,
Here, · i is the average over the N observed subframe images
                                                                                which involves the product of three phase values (Eq. (4)). It has
that incorporate independent realizations of atmospheric distor-
                                                                                been shown that both implemented algorithms can be equivalent
tions. The two-dimensional, spatial frequency shift vector δ can
                                                                                (Ayers et al. 1988). However, the Knox-Thompson algorithm
have a magnitude of up to the seeing limit in the Fourier do-
                                                                                is sensitive to alignment errors of the speckle images, whereas
main, r0 /λ, where λ denotes the observed wavelength and r0 is
                                                                                triple correlation algorithms do not suffer from this because of
the Fried parameter describing the prevailing atmospheric con-
                                                                                the phase closure relation inherent to the bispectrum. In bad see-
ditions. For large N, it can then be shown that the atmospheric
                                                                                ing conditions, this leads to a higher reconstruction error when
transfer function associated with the KT average cross-spectrum,
                                                                                using the extended Know-Thompson algorithm and a better per-
KT T F( f ) = S i ( f ) S i∗ ( f − δ) i ,                                (3)    formance of the speckle masking algorithm.

remains finite up to the diffraction limit, D/λ, with D being
the telescope pupil diameter. In addition, it is a real entity                  2.2. Amplitude reconstruction
merely scaling the Fourier amplitudes (Knox & Thompson 1974;                    The KISIP package reconstructs the object’s Fourier amplitudes
von der Lühe 1988). Thus, the extraction of the object’s Fourier                using the well-known method of Labeyrie (1970). With
phases O( f )/|O( f )| becomes possible from Eq. (2) by use of a
recursive or iterative algorithm. The incorporated algorithm ex-                 |Ii ( f )|2 i = |O( f )|2 |S i ( f )|2   i                       (6)
tends the original idea of Knox and Thompson by using more                      the object’s spatial power spectrum |O( f )| becomes accessible
                                                                                                                                  2
than two (linear independent) vectors δ. The extension is de-                   if the speckle transfer function (STF)
tailed in von der Lühe (1993), and Mikurda & von der Lühe
(2006).                                                                         S T F( f ) = |S i ( f )|2   i                                     (7)
                                F. Wöger et al.: Speckle interferometry with adaptive optics corrected solar data                             377




Fig. 2. Convergence properties of the two implemented algorithms. Upper row: KT algorithm, lower row: IWLS algorithm. Columns from left to
right: r0 = 5, 7, 10, 20 cm. Note that a panel of the IWLS algorithm corresponds to a subpanel of a KT panel. Shown is the residual phase variance
per pixel in the Fourier domain.


is known. Due to the lack of possibility to simultaneously ob-            present, there is a need for post-processing the data to achieve the
serve a reference point source in the sky when observing the              diffraction limit of the telescope as often as possible. To achieve
sun, the Fourier amplitudes need to be calibrated with model              this goal, an analysis has been performed to analyze several im-
STFs, the accuracy of which is vital to the photometry of the             portant aspects of the KISIP code.
final reconstruction. In order to chose the correct model func-
tion, the value of Fried’s parameter r0 , a measure for the strength
of atmospheric turbulence, needs to be well known. When solar             3.1. Code scalability
data is reconstructed, the most common way to estimate r0 is
                                                                          The KISIP code has been tested on various combinations of plat-
to compute the spectral ratio from the observed data itself. This
                                                                          forms and operating systems, demonstrating its scalability with
method was suggested by von der Lühe (1984) and is generally              an increasing number of computational nodes and platform in-
used by all solar speckle reconstruction algorithms. We have im-
                                                                          dependence. For the tests, we used a data burst with 100 images
proved the method for the estimation of r0 originally described
                                                                          consisting of 1024 ×1024 pixels. This data set was reconstructed
in von der Lühe (1984) to achieve a higher accuracy especially            with either 212 992 cross-spectrum (KT) or 221 320 bispectrum
in situations where the spectral ratio is not well defined. A di-
                                                                          (IWLS) values and 30 iterations per 128 × 128 pixel subfield.
rect, iterative fit of model spectral ratios to the measured data,
                                                                          We recorded the time to perform the reconstruction using an in-
using the squared differences between model and data as a met-             creasing number of computational nodes up to the maximum.
ric, increases reliability because more data points are used in the
fit. The model functions are precomputed and accessible dur-                   We present in Fig. 1, the results from tests run on a SuSE
ing the process via a lookup table. When an AO system was                 Enterprise Linux 10 cluster with 23 computational nodes plus
used for observations, the models need to be adjusted for the             one master node for job administration. This facility is in-
AO’s performance and atmospheric anisoplanatism (Wöger &                  stalled at the Kiepenheuer-Institut für Sonnenphysik. Each of
von der Lühe 2007). To correct for anisoplanatism, amplitudes             the 24 nodes is equipped with 2 Intel Xeon CPU 5160 with
are calibrated separately within each subfield: the spectral ratio         3.00 GHz clock speed and 4 GB of random access memory
delivers the appropriate value of r0 and, for AO-corrected data,          (RAM). The employed CPUs have 2 cores leading to a total
the estimated distance from the AO reference point. The details           number of 92 usable processing units – the master node is usu-
are described in Wöger & von der Lühe (2007). Using this in-              ally not involved in computations. Each computer was connected
formation, the photometric properties of the observed object can          to the main node via Infiniband fibers. As expected, the IWLS
be reconstructed to the highest accuracy possible.                        is slower than the KT algorithm because of the more involved
                                                                          computation. Additionally, Fig. 1 demonstrates that the code be-
                                                                          haves linearly with an increasing number of nodes: the compu-
                                                                          tational time decreases with the inverse of the number of nodes.
3. Reconstruction performance
                                                                          However, there is a saturation in reconstruction time at around
We are interested in the possibility of using speckle interferom-         22 s using both algorithms on this platform.
etry in real-time applications. Future ground-based solar tele-               The saturation is an important issue that needs to be paid
scopes will have have apertures of 1.5 m and more (Volkmer                close attention to when designing a platform that is supposed
et al. 2006; Denker et al. 2006; Wagner et al. 2006) and will             to achieve (near) real-time reconstruction performance. The sat-
be equipped with AO systems to acquire high-resolution obser-             uration is due to latency between the processors, be it because
vations. As the correction of an AO system can only be partial            of restricted interconnect bandwidth between the computation
and its performance is dependent on the atmospheric conditions            nodes or because of a slow processor speed. Another reason for
378                            F. Wöger et al.: Speckle interferometry with adaptive optics corrected solar data




Fig. 3. Top: deconvolved image of the quiet Sun region near disk center observed with Hinode on April 18th, 2007, at 15:30:30 UT. Bottom: the
same region observed with the Dunn Solar Telescope (DST); the data was post-processed using KISIP. Images are shown using the same intensity
scale.
                                        F. Wöger et al.: Speckle interferometry with adaptive optics corrected solar data                        379




Fig. 4. Close-up region of the region indicated in Fig. 3. Left: deconvolved Hinode image. Right: reconstructed DST image. Images are shown
using the same intensity scale.


saturation is the overhead in the code that distributes the data                  computation. The penalty is longer computational time, as men-
to the computation nodes. Thus, an ideal system would pro-                        tioned before. Nevertheless, greater than 30 iterations (or even
vide several multi-core processors that are connected with a                      less in case of the IWLS algorithm) in combination with ap-
fast system bus, which is the current trend in processor devel-                   proximately 250 000 evaluated cross- or bi-spectrum values for a
opment and high-performance computing. Nevertheless, already                      128×128 pixel subfield do not lead to a significant further change
today a system such as the one tested above would provide near                    in the reconstructed phase, which allows for the minimization of
real-time performance for a camera that reads out and stores a                    computational time by optimizing the reconstruction parameters.
1024 × 1024 pixel frame at an effective rate of 5 frames per sec-                  This fact is important with respect to real-time reconstruction of
ond.                                                                              speckle data.

3.2. Convergence properties                                                       3.3. Photometric accuracy
In addition to the code’s scalability, we analyzed its conver-
gence properties with synthetically distorted data cubes, which                   To test the accuracy of the phase reconstruction, we compare
have been aberrated using phase screens that correspond to                        speckle reconstructed data taken at the DST of the National Solar
atmospheric conditions that are similar to values of r0 =                         Observatory with data observed simultaneously with the SOT
5, 7, 10, 20 cm. The resulting 4 data cubes had 100 images of size                instrument on the Hinode satellite. The data were observed at
256 × 256 pixels. They are the same sets as used in the study of                  15:30:30 UT on April 18th, 2007, with both facilities using very
Mikurda & von der Lühe (2006). We were interested in the num-                     similar interference filters of 1 nm FWHM in the Fraunhofer G-
ber of cross- and bi-spectrum values as well as iterations needed                 band at 430.5 nm. A region of quiet solar granulation near Sun
for a satisfactory convergence of the iterative phase reconstruc-                 center was the target of the observations, the Fried parameter
tion for a subfield size of 128 × 128 pixels. For each iteration,                  was estimated to be r0 ≈ 7 cm, corresponding to average seeing.
we compute the property                                                           The data observed at the DST and at Hinode have been calibrated
                                                                                  using standard flatfielding procedures.
          1                                                                           The DST speckle burst was reconstructed using the IWLS
κ( j) =           (Φ j ( f ) − Φ j−1 ( f ))2 ,                             (8)
          M                                                                       algorithm with a subfield size of 128 × 128 pixels, which corre-
              f
                                                                                  sponds to approximately 7 . It is important for the photometric
where j represents the jth iteration step and M is the total num-                 accuracy that the subfield size be chosen based on the size of
ber of evaluated frequency points f . The quantity κ( j) measures                 the isoplanatic patch, or smaller. However, smaller subfields than
the squared difference from the current iteration step from the                    128 × 128 pixel subfields are not recommended because numer-
previous, which is an indicator for convergence.                                  ical issues during the Fried parameter estimation could arise.
    Figure 2 shows the results of a detailed analysis of the im-                       The Hinode image was deconvolved using a point spread
plemented algorithms, focusing on the convergence properties                      function that was computed from the measured aberrations of the
in dependence of atmospheric conditions and number of evalu-                      SOT main mirror (Suematsu et al. 2008). Is addition, a Wiener
ated cross- and bispectrum values. As can be seen, for both the                   filter was applied, with a noise estimate derived from the power
KT and the IWLS algorithms, these parameters are of impor-                        at frequencies that are higher than the theoretical diffraction cut-
tance for convergence. With less severe atmospheric conditions,                   off frequency. The deconvolution is necessary to make the in-
both algorithms converge faster as the signal-to-noise ratio in                   formation content of the Hinode image comparable to that of the
the images increases with increasing values of r0 . Generally, the                speckle interferometric reconstruction. It is successful up to 80%
KT algorithm seems to converge more slowly than the IWLS                          of the diffraction limit of the telescope. Beyond those spatial fre-
algorithm, which is likely the result of the additional informa-                  quencies, the employed Wiener filter cuts off the signal due to a
tion that is used in the averaging process of the bispectrum                      poor signal-to-noise ratio.
380                             F. Wöger et al.: Speckle interferometry with adaptive optics corrected solar data




Fig. 5. Left: intensity histograms for the Hinode (red) and the DST image (black) as shown in Fig. 3. Right: azimuthally-integrated, spatial-power
spectra of the Hinode (red) and the DST image (black).


    After reduction and alignment of the DST data to that of                  Another measure for the similarity of images are the “image
Hinode, the overall overlap in the images is 912 × 912 pixels,            distance” metrics defined in Mikurda & von der Lühe (2006),
corresponding to a field of view of almost 50 . Figure 3 demon-            here restated for convenience.
strates that the speckle algorithm is capable of reconstructing the
same structures seen by a telescope that is not hampered by at-                  1
                                                                          D2 =           (IDST (x) − IHin (x))2                              (10)
mospheric turbulence. The minor differences in fine structure of                   A   x
the images arise mainly from the fact that the speckle burst of
the DST spans approximately 20 s, as opposed to the single ex-            and
posure of the Hinode satellite. Thus, the data is only in approx-
imation simultaneous and some differences can be attributed to                    1
                                                                          E2 =           (IDST (x) − a − bIHin (x))2 ,                       (11)
the evolution of the granulation. However, as the spatial corre-                 A   x
lation time of the solar granulation is approximately 5 min, the
effect is small.                                                           where A is the area of the images. We have evaluated the eu-
                                                                          clidean image distance to be D2 = 0.00368925, and E 2 =
   The photometric differences in the images are evaluated in
                                                                          0.00316514. The values of a and b were computed from a lin-
several ways. We calculate the contrast of an image I with
                                                                          ear regression analysis of the scatter of reconstructed and Hinode
                                                                          image intensity. Again, the values demonstrate a good agreement
                                                                          of the two images.
C I = σI / I ,                                                      (9)
                                                                          4. Conclusions
where σI is the standard deviation of the mean value I . In the           We have presented the KISIP software package, which is capa-
speckle reconstructed DST image, this value is CDST = 15.1%,              ble of reconstructing solar speckle interferometric data observed
which is close to the value CHin = 16.3% in the deconvolved               using an AO system. The program is optimized to run in multi-
Hinode image. To analyze this further, we computed histograms             processor environments to make use of parallel computing capa-
of the images’ intensities with a binsize of 0.01 in normalized           bilities. While the reconstruction algorithms are based on well-
intensity. The histograms, shown in Fig. 5 (left), are very sim-          known principles they had to be adapted for use with solar data,
ilar and indicate the similarity of the intensity distribution. The       e.g., the amplitude calibration and estimation of the Fried pa-
differences can be attributed to certain spatial scales using the ra-      rameter r0 . The program scales well with an increasing number
dially integrated, spatial power spectra shown for both images in         of nodes and shows good convergence properties in every sit-
Fig. 5 (right). The abscissa is normalized to the theoretical spa-        uation tested. We have shown that using this program, with a
tial cutoff of Hinode’s SOT. Both curves show a striking similar-          high performance computing cluster, can lead to near real-time
ity, demonstrating the accuracy of the amplitude reconstruction           reconstruction performance.
discussed above. We attribute the deviations noticeable at nor-                The reconstruction accuracy has been demonstrated by com-
malized frequency 0.5 and above to the reduction of the signal-           parison to data observed co-spatially and co-temporally with the
to-noise ratio caused by anisoplanatism. While anisoplanatism             Hinode satellite. We have presented evidence that not only the
has been accounted for in the implemented model transfer func-            fine structure in ground-based data can be reconstructed well
tions, the signal-to-nose ratio in the fields furthest away from           with this computer program, but also that high photometric accu-
the structures that were used as reference for the AO correction          racy can be achieved, even when the data that was obtained with
(“lock-point”) is lower. This can lead to lower phase reconstruc-         an AO system. This has been achieved by implementing new
tion performance in those parts of the observed field, and thus            models for the object’s Fourier amplitude calibration. Satellite
reduced contrast contributions from higher spatial frequencies.           and ground-based data match very well.
                                    F. Wöger et al.: Speckle interferometry with adaptive optics corrected solar data                                           381

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